Bispectral mode decomposition of nonlinear flows
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ORIGINAL PAPER
Bispectral mode decomposition of nonlinear flows Oliver T. Schmidt
Received: 12 June 2020 / Accepted: 16 October 2020 © Springer Nature B.V. 2020
Abstract Triadic interactions are the fundamental mechanism of energy transfer in fluid flows. This work introduces bispectral mode decomposition as a direct means of educing flow structures that are associated with triadic interactions from experimental or numerical data. Triadic interactions are characterized by quadratic phase coupling which can be detected by the bispectrum. The proposed method maximizes an integral measure of this third-order statistic to compute modes associated with frequency triads, as well as a mode bispectrum that identifies resonant three-wave interactions. Unlike the classical bispectrum, the decomposition establishes a causal relationship between the three frequency components of a triad. This permits the distinction of sum- and difference-interactions, and the computation of interaction maps that indicate regions of nonlinear coupling. Three examples highlight different aspects of the method. Cascading triads and their regions of interaction are educed from direct numerical simulation data of laminar cylinder flow. It is further demonstrated that linear instability mechanisms that attain an appreciable amplitude are revealed indirectly by their difference-self-interactions. Applicability to turbulent flows and noise-rejection is demonstrated on particle image velocimetry data of a massively separated wake. The generation of sub- and ultra-harmonics in large O. T. Schmidt (B) Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA e-mail: [email protected]
eddy simulation data of a transitional jet is explained by extending the method to cross-bispectral information. Keywords Nonlinear flows · Triadic interactions · Modal decompositions · Proper orthogonal decomposition
1 Introduction Triadic interactions result from the quadratic nonlinearity of the Navier–Stokes equations. They are the fundamental mechanism of energy transfer in fluid flows and manifest, in Fourier space, as triplets of three wavenumber vectors, {k j , kk , kl }, or frequencies, { f j , f k , fl }, that sum to zero: k j ± kk ± kl = 0,
(1a)
f j ± f k ± fl = 0.
(1b)
For clarity, we denote by {·} multiplets of frequency or wavenumber, and by (·) index multiplets. The zero-sum condition implies that triads form triangles in wavenumber- and frequency-space. A way to conceptually visualize these three-wave interactions is presented in Fig. 1. Since the early work of Phillips [43] on weak resonant interactions of gravity waves on the surface of deep water, interaction theory has vastly improved our understanding of nonlinear wave phenomena. The turbulent cascade, which describes the transfer of energy from large to small scales of motion,
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O. T. Schmidt (a)
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Fig. 1 Illustration of typical frequency triads: a generic suminteraction; b generic difference-interaction; c mean-flow deformatio
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